LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A tournament vertex having an in-neighbour and an out-neighbour on a directed cycle can be inserted into that cycle

Statement

Let $C$ be a directed cycle in a tournament, and let $x$ be a vertex outside $C$ . If some vertex of $C$ sends an arc to $x$ and $x$ sends an arc to some vertex of $C$ , then the tournament has a directed cycle whose vertex set is $V (C) \cup {x}$ .

Facts & Assumptions

Given: A directed cycle $C = (c_{1}, c_{2}, \dots, c_{k}, c_{1})$ and an outside vertex $x$ satisfying the two incidence hypotheses.

[F1]

Between $x$ and each $c_{i}$ , exactly one arc direction occurs (A tournament is an orientation of a complete finite graph).

[F2]

A directed cycle follows its cyclic sequence of arcs and has distinct vertices before returning to its start (Directed walks, trails, paths and cycles, and strong connectivity).

Proof

technique · constructive

1.1

Mark a cycle vertex $c_{i}$ with $I$ when $c_{i} x$ is an arc and with $O$ when $x c_{i}$ is an arc. Both marks occur by hypothesis, and [F1] gives exactly one mark at each cycle vertex. Therefore somewhere around the cyclic order an $I$ -marked vertex $c_{i}$ is followed by an $O$ -marked vertex $c_{i + 1}$ .

givenF1F2

2.1

Replace the cycle arc $c_{i} c_{i + 1}$ by the two arcs $c_{i} x$ and $x c_{i + 1}$ . All other cycle arcs remain unchanged, and $x$ is new, so the resulting closed directed walk is a directed cycle on $V (C) \cup {x}$ .

step 1.1F2construct

3.1

This explicitly constructs the required enlarged directed cycle.

step 2.1discharge-construct∎

Depends on

Used by

Camion's theorem: every strongly connected tournament on at least three vertices has a directed Hamilton cycle Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Encyclopedia of Mathematics, Tournament (standard reference, not scraped)
Robin Wilson, Introduction to Graph Theory (standard reference, not scraped)